Term Rewriting System R:
[x, y]
g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(f(x), y) -> H(x, y)
H(x, y) -> G(x, f(y))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

H(x, y) -> G(x, f(y))
G(f(x), y) -> H(x, y)


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))





The following dependency pair can be strictly oriented:

G(f(x), y) -> H(x, y)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> x1
f(x1) -> f(x1)
H(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:

H(x, y) -> G(x, f(y))


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes